Finite Group Databasebeta

← #6374062

Subgroups — #6374062

Order 512.

1476 conjugacy classes of subgroups.

The subgroup lattice has 323 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34644C211yesmaximal, characteristic
2#21716C212yesmaximal, complemented
3#21715C212yesmaximal, complemented
4#21677C212yesmaximal, complemented
5#21676C212yesmaximal, complemented
6#21603C214yesmaximal
7#21601C214yesmaximal
8#21352C212yesmaximal
9#21342C214yesmaximal, complemented
10#20811C211yesmaximal, characteristic
11#20516C214yesmaximal, complemented
12#20476C212yesmaximal
13C4×#405C211yesmaximal, characteristic
14C2×#446C2214yes—
15C2×C8×Q8C2214yes—
16C2×#432C2212yes—
17C2×#405C2212yes—
18C2×#405C2212yes—
19C2×C4×M16C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2667C414yes—
22#2666C418yes—
23#2665C414yes—
24#2402C2212yes—
25#2399C2212yes—
26#2396C2214yes—
27#2295C2214yes—
28#2283C2212yes—
29C4×#117C2211yescharacteristic
30#1985C2214yes—
31#1985C2214yes—
32#1940C2214yes—
33#1940C2214yes—
34#1940C2214yes—
35#1940C2214yes—
36#1917C2218yes—
37#1917C2218yes—
38#1851C2218yes—
39#1843C2212yes—
40#1832C2212yes—
41#1798C2214yes—
42#1771C2218yes—
43#1770C2214yes—
44#1770C2214yes—
45C8×C4⋊C4C2218yes—
46C8×C4⋊C4C2214yes—
47C8×C4⋊C4C2214yes—
48#1768C2218yes—
49#1767C2214yes—
50#1752C2214yes—
51C8×C22⋊C4C2214yes—
52#1746C2214yes—
53#1726C2214yes—
54#1726C2211yescharacteristic
55#1726C2211yescharacteristic
56#1726C2211yescharacteristic
57#1726C2211yescharacteristic
58C4×#97C2212yes—
59C42×C8C2212yes—
60C22×M16C2312yes—
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×C4×Q8C2314yes—
64C2×#117C2312yes—
65C22×C42C2311yescharacteristic
66#446—216no—
67C8×Q8—216no—
68#432—28no—
69C2×C4⋊C8C2314yes—
70C2×C4⋊C8C2314yes—
71C2×C4⋊C8C2314yes—
72C2×C4⋊C8C2318yes—
73C2×C4⋊C8C2318yes—
74C2×C22⋊C8C2314yes—
75#405—28no—
76#405—28no—
77C4×M16—24no—
78C2×#97C2311yescharacteristic
79C2×#97C2311yescharacteristic
80C2×#97C2312yes—
81C2×#97C2312yes—
82C2×#97C2312yes—
83C2×#97C2312yes—
84C2×#97C2314yes—
85C2×#97C2318yes—
86C2×C4×C8Q818yescomplemented
87C2×C4×C8C2312yes—
88C2×C4×C8C2312yes—
89C2×C4×C8C2312yes—
90C2×C4×C8C2318yes—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2312yes—
96C2×C4×C8C2312yes—
97#400C2×C414yes—
98#400C2×C418yes—
99#399C2×C414yes—
100#398C2×C418yes—
101#398C2×C418yes—
102#398C2×C418yes—
103#396C2×C418yes—
104#395C2×C414yes—
105#395C2×C418yes—
106#393C2×C414yes—
107#391C2×C418yes—
108#389C2×C418yes—
109#387C2×C418yes—
110#387C2×C418yes—
111#383C2×C414yes—
112#382C2×C414yes—
113#381C2×C412yes—
114#381C2×C412yes—
115#380C2×C414yes—
116C4×C4⋊C4C2×C414yes—
117C4×C4⋊C4C2314yes—
118C4×C4⋊C4C2318yes—
119C4×C4⋊C4C2314yes—
120C4×C22⋊C4C2×C414yes—
121C4×C22⋊C4C2314yes—
122#376C2311yescharacteristic
123#376C2311yescharacteristic
124#376C2314yes—
125C43C2312yes—
126#336C2318yes—
127#336C2318yes—
128#336C2318yes—
129#336C2318yes—
130#336C2318yes—
131C22×Q8C22×C414yes—
132C23×C4C22×C412yes—
133C23×C4C2411yescharacteristic
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8C2×Q814yes—
140C22×C8C2×Q814yes—
141C22×C8C2418yes—
142C22×C8C2411yescharacteristic
143C22×C8C2411yescharacteristic
144C22×C8C2412yes—
145C22×C8C2412yes—
146C22×C8C2411yescharacteristic
147C22×C8C2411yescharacteristic
148C4×Q8—28no—
149C4×Q8—28no—
150#117—24no—
151#117—24no—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C418yes—
154C2×C4⋊C4C22×C414yes—
155C2×C4⋊C4C22×C418yes—
156C2×C4⋊C4C22×C418yes—
157C2×C4⋊C4C22×C418yes—
158C2×C4⋊C4C22×C418yes—
159C2×C22⋊C4C22×C414yes—
160C2×C22⋊C4C22×C414yes—
161C2×C22⋊C4C22×C414yes—
162C2×C42C22×C418yes—
163C2×C42C22×C412yes—
164C2×C42C2×Q814yes—
165C2×C42#4114yes—
166C2×C42C2412yes—
167C2×C42C2418yes—
168C2×C42C2411yescharacteristic
169C2×C42C2411yescharacteristic
170C2×C42C2412yes—
171C4⋊C8—216no—
172C4⋊C8—216no—
173C4⋊C8—28no—
174C4⋊C8—28no—
175C4⋊C8—28no—
176C22⋊C8—28no—
177#97—24no—
178#97—24no—
179#97—216no—
180#97—22no—
181#97—22no—
182#97—24no—
183#97—24no—
184#97—28no—
185C4×C8—24no—
186C4×C8—24no—
187C4×C8—216no—
188C4×C8—24no—
189C4×C8C2×Q814yes—
190C4×C8C2×Q814yes—
191C4×C8C2×Q814yes—
192C4×C8C2×Q814yes—
193C4×C8#4118yes—
194C4×C8#4118yes—
195#95C22×C414yes—
196#95C22×C414yes—
197#95C22×C418yes—
198#95C22×C414yes—
199#95C22×C414yes—
200#95C22×C418yes—
201#95C22×C414yes—
202#95C22×C414yes—
203#95C22×C414yes—
204#95C22×C414yes—
205#95C22×C418yes—
206#95C22×C418yes—
207C24C23×C411yescharacteristic
208C2×Q8—28no—
209C2×Q8—48no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—44no—
213C22×C4C4×Q818yes—
214C22×C4C23×C418yes—
215C22×C4C23×C414yes—
216C22×C4C23×C418yes—
217C22×C4C23×C412yes—
218C22×C4C23×C411yescharacteristic
219C22×C4C23×C412yes—
220C22×C4C23×C414yes—
221C22×C4C22×Q812yes—
222C22×C4C2×#4112yes—
223C22×C4C2511yescharacteristic
224M16—216no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—28no—
232C2×C8—28no—
233C2×C8—28no—
234C2×C8—28no—
235C2×C8—28no—
236C2×C8C22×Q812yes—
237C2×C8C22×Q812yes—
238C2×C8C22×Q812yes—
239C2×C8C22×Q812yes—
240C2×C8C2×#4114yes—
241C2×C8C2×#4114yes—
242C2×C8C2×#4114yes—
243C2×C8C2×#4114yes—
244C4⋊C4—28no—
245C4⋊C4—28no—
246C4⋊C4—216no—
247C4⋊C4—48no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C22⋊C4—28no—
252C22⋊C4—44no—
253C22⋊C4—44no—
254C42—22no—
255C42—22no—
256C42—28no—
257C42—28no—
258C42—48no—
259C42C4×Q818yes—
260C42C22×Q812yes—
261C42C22×Q812yes—
262C42C2×#4112yes—
263C42C2×#4112yes—
264C42C2×#4114yes—
265C42C2×#4114yes—
266C23—22no—
267C23—24no—
268C23C2×C4×Q812yes—
269C23C24×C411yescharacteristic
270Q8—416no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—28no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—28no—
280C2×C4—28no—
281C2×C4—28no—
282C2×C4—28no—
283C2×C4—28no—
284C2×C4—28no—
285C2×C4—28no—
286C2×C4—48no—
287C2×C4C2×C4×Q812yes—
288C2×C4C2×C4×Q814yes—
289C2×C4C2×C4×Q814yes—
290C2×C4C4×#4118yes—
291C2×C4C23×Q811yescharacteristic
292C2×C4C22×#4112yes—
293C2×C4C22×#4112yes—
294C2×C4C22×#4111yescharacteristic
295C8—22no—
296C8—22no—
297C8—24no—
298C8—24no—
299C8—48no—
300C8#53314yes—
301C8#53314yes—
302C22—24no—
303C22—22no—
304C22—44no—
305C22C22×C4×Q811yescharacteristic
306C22C2×C4×#4112yes—
307C22#357314yes—
308C4—24no—
309C4—24no—
310C4—28no—
311C4—22no—
312C4—44no—
313C4—48no—
314C4—48no—
315C4C2×#53311yescharacteristic
316C4C2×#53311yescharacteristic
317C4#344412yes—
318C2—42no—
319C2C4×#53311yescharacteristic
320C2#6019112yes—
321C2#6020614yes—
subgroupG/H
centerC22×C8C24
commutator, socleC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21716C2semidirect
3. #21715C2semidirect
4. #21677C2semidirect
5. #21676C2semidirect
9. #21342C2semidirect
11. #20516C2semidirect
86. C2×C4×C8Q8semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#346442
#217162
#217152
#216772
#216762
#216032
#216012
#213522
#213422
#208112
#205162
#204762
C4×#4052